derived category - determinant of a perfect complex

As I understand the construction of the determinant of a perfect complex, this definition is quite straightforward, following from the fact that in a short exact sequence, say
$$ 0rightarrow Srightarrow Erightarrow Qrightarrow 0$$
defining the determinant of the sequence to be the alternating tensor is the canonical way to make it isomorphic to $mathbb{1}$.

Also, I think good references to this may be the original paper by Knudsen-Mumford, a book by Kato, and also a paper by Kings which are listed below:

Finn Faye Knudsen and David Mumford, The projectivity of the moduli space of stable curves. I. Preliminaries on ''det'' and ''Div'' (pdf). The part about determinants appears in Chapter I, but note that there is a typo defining the determinant, namely in the map of the transposition of tensor product, there should be $alphacdotbeta$ instead the sum of these two as a power of $-1$;

Guido Kings, An introduction to the equivariant Tamagawa number conjecture: the relation to the Birch-Swinnerton-Dyer conjecture (pdf)

There is a part about determinants in lecture 1 section 5, where there are not a lot of details but it provides a good view towards the construction of determinant.

Kazuya Kato, Lectures on the approach to Iwasawa theory for Hasse-Weil L-functions via $B_{dR}$, part I Springer LNM 1553 pp 50-163 (doi:10.1007/BFb0084729), which mentions determinant in 2.1.